Multi-channel grating structures are typically written into photosensitive waveguides. The grating structure comprises a refractive index variations created in the photosensitive waveguide, which in turn determine the optical characteristics such as the reflection and transmission characteristics of the resulting grating structure.
The envelope of the refractive index variation for a multi-channel grating structure is typically created by applying a sampling, i.e. periodic, function to a given single-channel grating design function. The single-channel grating profile is then typically an envelope of the refrcative index variations achieved by exposing the photosensitive waveguide through a suitable phasemask or using other interferometric techniques.
In a vast majority of previously reported work on multi-channel gratings, a so-called Sinc-sampled design has been used. For the Sinc-sampling approach, an N-channel grating design can be obtained by a direct in-phase summation of N identical seeding gratings [with κ(z)—amplitude grating amplitude, θ(z)—grating phase] equally spaced in the frequency space:
                                                                        ∑                                  l                  =                  1                                N                            ⁢                              κ                ⁢                                                                  ⁢                                  ⅇ                                      ⅈ                    ⁢                                                                                  [                                                                                            K                          0                                                ⁢                        z                                            +                      θ                      +                                                                        (                                                                                    2                              ⁢                                                                                                                          ⁢                              l                                                        -                            N                            -                            1                                                    )                                                ⁢                        Δ                        ⁢                                                                                                  ⁢                        κ                        ⁢                                                                                                  ⁢                                                  z                          /                          2                                                                                      ]                                                                        =                          κ              ⁢                                                          ⁢                              Q                                  Sin                  ⁢                                                                          ⁢                  c                                            ⁢                              ⅇ                                  ⅈ                  ⁢                                                                          ⁢                                      (                                                                                            K                          0                                                ⁢                        z                                            +                      θ                                        )                                                                                ,                                          ⁢          where                ⁢                                  ⁢                                                                              Q                                      Sin                    ⁢                                                                                  ⁢                    c                                                  =                                ⁢                                                      ∑                                          l                      =                      1                                        N                                    ⁢                                      cos                    ⁡                                          [                                                                        (                                                                                    2                              ⁢                              l                                                        -                            N                            -                            1                                                    )                                                ⁢                        Δ                        ⁢                                                                                                  ⁢                        k                        ⁢                                                                                                  ⁢                                                  z                          /                          2                                                                    ]                                                                                                                                                                =                                    ⁢                                      N                    ⁢                                                                  ∑                                                  n                          =                                                      -                            ∞                                                                                                    +                          ∞                                                                    ⁢                                              sin                        ⁢                                                                                                  ⁢                                                  c                          ⁡                                                      [                                                                                          N                                ⁡                                                                  (                                                                                                            Δ                                      ⁢                                                                                                                                                          ⁢                                      k                                      ⁢                                                                                                                                                          ⁢                                      z                                                                        -                                                                          2                                      ⁢                                                                                                                                                          ⁢                                      π                                      ⁢                                                                                                                                                          ⁢                                      n                                                                                                        )                                                                                            /                              2                                                        ]                                                                                                                                              ,                                                    ⁢                                  ⁢                                            sin              ⁢                                                          ⁢                              c                ⁡                                  (                  x                  )                                                      ≡                                          sin                ⁡                                  (                  x                  )                                            /              x                                ,                                    (        1        )            
and Δk is the channel spacing.
This design will be referred to as “in-phase” grating design herein after by the applicant.
An example of this design is shown in FIGS. 4(c) and (d) with corresponding spectral characteristics as shown in FIGS. 4(a) and (b). The maximum value of the refractive index change required to implement this multi-channel grating design is given by a simple expression:ΔnN(max)=NΔns,  (2)where Δns is the maximum refractive index change required for the single seeding grating. Since any photosensitive fiber used to fabricate Bragg gratings has material limits of the maximum achievable photoinduced refractive index change ΔnN this represents a limitation on the maximum number of channels that can be recorded in a given fiber. Thus it is highly desirable to reduce a required ΔnN as much as possible. Also it is easy to see that in the chosen example (see FIG. 4) the substantial deviations from the desired (square-like in transmission; linear in group delay) spectral characteristics are present. It has been found that such deviations are always present, albeit to different degrees, if a strictly periodic sampling function approach is used.
In another approach, one may solve an inverse scattering problem for a multi-channel grating directly (i.e. without calculating a single channel seeding profile first and then applying a sampling function). An example is given in FIGS. 5(a)–(d). As can be seen, the spectral characteristics are substanially perfect, but ΔnN is poorly optimised.
At least preferred embodiments of the present invention seek to provide an alternative multi-channel grating design in which (i) the maximum refractive index change required as a function of the number of channels is reduced when compared with the prior art grating designs discussed above and (ii) the resulting multi-channel gratings exhibit substantially a desired shape, e.g. square-like shape, in their transmission co-efficient characteristics.